Could not include topos theory - contents
A coverage (resp. Grothendieck topology, resp. Grothendieck pretopology) defining a site is called subcanonical if all representable presheaves on this site are sheaves. Of course, a subcanonical site is one whose coverage is subcanonical.
The term “subcanonical” comes about because the largest coverage for which the representables are sheaves is called the canonical coverage, and the subcanonical coverages are precisely the “sub-coverages” of the canonical one.
An alternate definition is that a Grothendieck coverage is subcanonical if and only if all of its covering sieves are effective-epimorphic, meaning that the morphisms in form a colimit cone under the diagram consisting of all morphisms between them over . To see this, first recall that if is a sieve, then a functor satisfies the sheaf axiom for if and only if
Interpreting this when is a representable functor , we obtain
But this says precisely that is effective-epimorphic, as defined above.
In fact, since the covering sieves in a subcanonical coverage must also satisfy pullback-stability, they must be not only effective-epimorphic but universally effective-epimorphic (meaning that any pullback of them is effective-epimorphic). It is then easy to see that the canonical coverage consists precisely of all the universally effective-epimorphic sieves.